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Properties

Label 4-546e2-1.1-c1e2-0-23
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s + 3·9-s − 2·12-s − 6·13-s + 16-s + 6·17-s − 2·23-s + 25-s + 4·27-s + 10·29-s − 3·36-s − 12·39-s − 2·43-s + 2·48-s − 49-s + 12·51-s + 6·52-s + 28·53-s − 6·61-s − 64-s − 6·68-s − 4·69-s + 2·75-s + 5·81-s + 20·87-s + 2·92-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 1.85·29-s − 1/2·36-s − 1.92·39-s − 0.304·43-s + 0.288·48-s − 1/7·49-s + 1.68·51-s + 0.832·52-s + 3.84·53-s − 0.768·61-s − 1/8·64-s − 0.727·68-s − 0.481·69-s + 0.230·75-s + 5/9·81-s + 2.14·87-s + 0.208·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.376411700\)
\(L(\frac12)\) \(\approx\) \(2.376411700\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.19.a_abl
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.59.a_da
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.61.g_fb
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.71.a_abq
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64424565668071911071373429551, −10.48578500125122597595345153564, −10.03045588248703020413097068827, −9.566490198455084588934479518599, −9.474835345348604022768159047682, −8.717431507189508441762219730357, −8.253158637349628081886891555853, −8.179094225529055222791246943536, −7.33807092747286015892275102407, −7.26015157601889847259167581047, −6.65867049724933434258730447769, −5.89706123978545200226975534533, −5.28186697409185948060013263027, −4.96054312587536855532952426758, −4.18675780044876769409520304301, −3.92534040678059442665584062182, −2.86521533116791941437839235681, −2.86349550330828889337040673671, −1.92143899985575659444802638883, −0.881998858320457016092041279158, 0.881998858320457016092041279158, 1.92143899985575659444802638883, 2.86349550330828889337040673671, 2.86521533116791941437839235681, 3.92534040678059442665584062182, 4.18675780044876769409520304301, 4.96054312587536855532952426758, 5.28186697409185948060013263027, 5.89706123978545200226975534533, 6.65867049724933434258730447769, 7.26015157601889847259167581047, 7.33807092747286015892275102407, 8.179094225529055222791246943536, 8.253158637349628081886891555853, 8.717431507189508441762219730357, 9.474835345348604022768159047682, 9.566490198455084588934479518599, 10.03045588248703020413097068827, 10.48578500125122597595345153564, 10.64424565668071911071373429551

Graph of the $Z$-function along the critical line